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During a Carnot cycle, why don't we consider the increase in entropy of surrounding by the energy given to surrounding via work done by gas piston? I mean we want to find the total entropy change then why don't we consider the entropy change due to this energy given to surrounding by work done through gas piston?

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The short answer is work does not transfer entropy. Entropy can only be transferred by heat. Work can generate entropy if it is irreversible work. But all work done in the Carnot Cycle is reversible, so no entropy is generated in the Carnot Cycle.

Hope this helps.

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When a Carnot cycle runs, the total universe has a net zero entropy change. This is by the definition of a reversible process.

However, since entropy is a state function, considering it gives us useful relations between the variables inside the Carnot system.

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We do consider the change in entropy of both the system and surroundings when we analyze the Carnot cycle. Since the Carnot engine is operating in a cycle, the change in entropy of the working fluid (the system) is always zero, irrespective of whether the engine is operating reversibly. If the engine is operating reversibly, the change in entropy of the surroundings is also zero.

In analyzing the reversible Carnot cycle, we have the option of considering the entropy change of either the working fluid or the surroundings (the two constant temperature reservoirs). Each has an entropy change of zero.

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